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Algebraic Structures and Groups Notes for GATE CS

Algebraic Structures and Groups notes for GATE CS: 38 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

algebraic structures and groups notes

Chapter Roadmap: Algebraic Structures and Groups

Orientation Algebraic Structures and Groups A structured route from binary operations to algebraic laws, then to monoids, groups, cyclicity, and subgroups. 1 Binary Operations and Algebraic Laws Custom operators, closure, commutativity, associativity, distributivity. Foundation 2 Monoids of Functions under Pointwise Operations Pointwise operations, identity elements on function spaces. Moderate 3 Groups from Set Operations and Direct Products Symmetric difference, power sets, Cartesian products of groups. High Yield 4 Group Properties, Cyclicity and Subgroups Generators, Lagrange's theorem, subgroup tests. High Yield

What is a Binary Operation?

Concept What is a Binary Operation? The core idea A binary operation is a rule that combines two elements from a set and returns exactly one element. The important part is not the symbol. The important part is whether the output stays inside the same set. Formal definition * : S \times S \to S It accepts an ordered pair (a, b) with a, b \in S . It gives exactly one result. We write the result as a * b = c . Golden rule: closure The result c must belong to S . If even one valid input pair produces an output outside S, the rule is not a binary operation on S . Example: on positive integers, 3 - 5 = - 2 . Since - 2 is not positive, subtraction is not closed on the positive integers.

Custom Operators in Exams

Concept Custom Operators in Exams Treat the symbol as a machine a ⋄ b = a + 2 b The left operand fills the a slot. The right operand fills the b slot. Do not assume the symbol behaves like + or \times . Order matters Left input first 3 ⋄ 4 = 3 + 2 (4) = 11 Swap the inputs 4 ⋄ 3 = 4 + 2 (3) = 10 Since 3 ⋄ 4 \neq = 4 ⋄ 3, this operation already shows that order can matter. That is the doorway into commutativity.

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Question 1

Level 1: Warm-up

Assertion (A): The number of self-inverse elements in Z_{2} \times Z_{3} \times Z_{4} is 4. Reason (R): An element (x, y, z) is self-inverse if 2 x \equiv 0 (mod 2), 2 y \equiv 0 (mod 3), and 2 z \equiv 0 (mod 4) .

Question 2

Level 1: Warm-up

Consider the following: Assertion (A): The index of a subgroup H in a group G of order 24, where ∣ H ∣ = 8, is 3. Reason (R): The index of H in G is calculated as the order of H divided by the order of G . Which option is correct?

Question 3

Level 1: Warm-up

Let a be an element of order 12 in an abelian group. Which of the following statements is TRUE?

Question 4

Level 1: Warm-up

Let A = {0, 1, 2, \dots} and F be the set of all functions from A to A . Under pointwise addition (f \oplus g) (n) = f (n) + g (n), how many functions in F possess an inverse?

Question 5

Level 1: Warm-up

Match the group with its self-inverse condition. List I: P) Z_{n} under addition Q) U (n) under multiplication List II: x^{2} \equiv 1 (mod n) 2 x \equiv 0 (mod n)

Question 6

Level 1: Warm-up

Consider the following: Assertion (A): The number of generators of the cyclic group Z_{12} is 4. Reason (R): An element k is a generator of Z_{n} if and only if g cd (k, n) = 1 . Which option is correct?

Question 7

Level 1: Warm-up

Let G = Z_{15} be the group of integers modulo 15 under addition. How many elements in G have an order of exactly 5?

Question 8

Level 1: Warm-up

Let A = {0, 1, 2, \dots} and F be all functions from A to A . Under pointwise addition (f \oplus g) (n) = f (n) + g (n), which task is IMPOSSIBLE?

Question 9

Level 1: Warm-up

Consider the following: Assertion (A): The operation * defined by a * b = a + 2 b on integers is associative. Reason (R): For associativity, we need (a * b) * c = a * (b * c) for all a, b, c . Which option is correct?

Question 10

Level 1: Warm-up

Let X be a set with 4 elements. The group (2^{X}, Δ) is isomorphic to a direct product of copies of Z_{2} . Which of the following is IMPOSSIBLE for this group?

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Algebraic Structures and Groups Notes for GATE CS

Algebraic Structures and Groups notes for GATE CS: 38 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Algebraic Structures and Groups

Orientation

Algebraic Structures and Groups

A structured route from binary operations to algebraic laws, then to monoids, groups, cyclicity, and subgroups.

Binary Operations and Algebraic Laws

Custom operators, closure, commutativity, associativity, distributivity.

Foundation

Monoids of Functions under Pointwise Operations

Pointwise operations, identity elements on function spaces.

Moderate

Groups from Set Operations and Direct Products

Symmetric difference, power sets, Cartesian products of groups.

High Yield

Group Properties, Cyclicity and Subgroups

Generators, Lagrange's theorem, subgroup tests.

High Yield

What is a Binary Operation?

Concept

What is a Binary Operation?

The core idea

A binary operation is a rule that combines two elements from a set and returns exactly one element.

The important part is not the symbol. The important part is whether the output stays inside the same set.

Formal definition

* : S \times S \to S

It accepts an ordered pair $(a, b)$ with $a, b \in S$ .
It gives exactly one result.
We write the result as $a * b = c$ .

Golden rule: closure The result

c

must belong to

S

. If even one valid input pair produces an output outside

S

, the rule is not a binary operation on

S

Example: on positive integers,

3 - 5 = - 2

. Since

- 2

is not positive, subtraction is not closed on the positive integers.

Custom Operators in Exams

Concept

Custom Operators in Exams

Treat the symbol as a machine

a ⋄ b = a + 2 b

The left operand fills the $a$ slot.
The right operand fills the $b$ slot.
Do not assume the symbol behaves like $+$ or $\times$ .

Order matters

Left input first

3 ⋄ 4 = 3 + 2 (4) = 11

Swap the inputs

4 ⋄ 3 = 4 + 2 (3) = 10

Since

3 ⋄ 4 \neq = 4 ⋄ 3

, this operation already shows that order can matter. That is the doorway into commutativity.

Commutativity: Does Order Matter?

Concept

Commutativity: Does Order Matter?

Definition

A binary operation $*$ on a set $S$ is commutative if $a * b = b * a for all a, b \in S .$

Start with the given expression:

a ⋄ b = a + 2 b

Swap the inputs:

b ⋄ a = b + 2 a

Compare:

a + 2 b

is not the same expression as

b + 2 a

for all values.

Conclusion The operation

⋄

is not commutative.

Addition and multiplication are commutative. Subtraction and division are not. Custom operators must be checked from their formulas.

Algebraic Structures and Groups: Solved Questions with Step-by-Step Explanations (10 Problems)

Question 1 · Engineering Mathematics MCQ

Assertion (A): The number of self-inverse elements in $Z_{2} \times Z_{3} \times Z_{4}$ is 4.

Reason (R): An element $(x, y, z)$ is self-inverse if $2 x \equiv 0 (mod 2)$ , $2 y \equiv 0 (mod 3)$ , and $2 z \equiv 0 (mod 4)$ .

A.
A is true but R is false.
B.
Both A and R are true but R is NOT the correct explanation of A.
C.
A is false but R is true.
D.
Both A and R are true and R is the correct explanation of A.

Correct Answer:

Step-by-Step Solution

Key idea: In a direct product, an element is self-inverse iff each component is self-inverse.

Step 1: In $Z_{n}$ under addition, the self-inverse condition is $2 x \equiv 0 (mod n)$ . This makes R true.

Step 2: For $Z_{2}$ , $2 x \equiv 0 (mod 2) ⟹ x \in {0, 1}$ (2 solutions).

Step 3: For $Z_{3}$ , $2 y \equiv 0 (mod 3) ⟹ y = 0$ (1 solution).

Step 4: For $Z_{4}$ , $2 z \equiv 0 (mod 4) ⟹ z \in {0, 2}$ (2 solutions).

Step 5: Total self-inverse elements = $2 \times 1 \times 2 = 4$ . This makes A true.

Step 6: R correctly provides the method to compute A, so R is the correct explanation.

Answer: D

Question 2 · Engineering Mathematics MCQ

Consider the following:

Assertion (A): The index of a subgroup $H$ in a group $G$ of order 24, where $∣ H ∣ = 8$ , is 3.

Reason (R): The index of $H$ in $G$ is calculated as the order of $H$ divided by the order of $G$ .

Which option is correct?

A.
Both A and R are true and R is the correct explanation of A
B.
A is true but R is false
C.
Both A and R are true but R is NOT the correct explanation of A
D.
A is false but R is true

Correct Answer:

Step-by-Step Solution

Key idea: The index of a subgroup is $∣ G ∣/∣ H ∣$ , not $∣ H ∣/∣ G ∣$ .

Step 1: Evaluate Assertion (A). The index of $H$ in $G$ is defined as $[G : H] = ∣ G ∣/∣ H ∣$ .

Step 2: Calculate the index: $24/8 = 3$ . Thus, Assertion (A) is true.

Step 3: Evaluate Reason (R). R states the index is $∣ H ∣/∣ G ∣$ . This is the reciprocal of the correct formula. Thus, Reason (R) is false.

Step 4: Conclude that A is true but R is false.

Answer: B

Question 3 · Engineering Mathematics MCQ

Let $a$ be an element of order 12 in an abelian group. Which of the following statements is TRUE?

A.
$o (a^{4}) = 4$
B.
$o (a^{5}) = 5$
C.
$o (a^{9}) = 9$
D.
$o (a^{8}) = 3$

Correct Answer:

Step-by-Step Solution

Key idea: The order of a power $a^{k}$ is given by $o (a^{k}) = \frac{o ( a )}{g c d ( o ( a ) , k )}$ .

Step 1: Recall the formula for the order of a power: $o (a^{k}) = \frac{n}{g c d ( n , k )}$ , where $n = o (a)$ .

Step 2: Here, $n = 12$ . We evaluate each option using the formula.

Step 3: For $a^{4}$ : $o (a^{4}) = 12/ g cd (12, 4) = 12/4 = 3$ . (Option A is false).

Step 4: For $a^{5}$ : $o (a^{5}) = 12/ g cd (12, 5) = 12/1 = 12$ . (Option B is false).

Step 5: For $a^{9}$ : $o (a^{9}) = 12/ g cd (12, 9) = 12/3 = 4$ . (Option C is false).

Step 6: For $a^{8}$ : $o (a^{8}) = 12/ g cd (12, 8) = 12/4 = 3$ . (Option D is true).

Answer: D

Question 4 · Engineering Mathematics MCQ

Let $A = {0, 1, 2, \dots}$ and $F$ be the set of all functions from $A$ to $A$ . Under pointwise addition $(f \oplus g) (n) = f (n) + g (n)$ , how many functions in $F$ possess an inverse?

A.
0
B.
1
C.
2
D.
infinitely many

Correct Answer:

Step-by-Step Solution

Key idea: For $f$ to have an inverse $g$ , we need $f \oplus g = e$ , where $e$ is the identity (zero function).

Step 1: Recall the identity. The identity function $e$ satisfies $f \oplus e = f$ for all $f$ . This means $f (n) + e (n) = f (n)$ , so $e (n) = 0$ for all $n$ . The identity is the zero function.

Step 2: Set up the inverse condition. For $f$ to have an inverse $g$ , we need $f \oplus g = e$ , i.e., $f (n) + g (n) = 0$ for all $n \in A$ .

Step 3: Analyze the constraint. Since $f (n) \in A$ and $g (n) \in A$ , both are non-negative integers. The sum $f (n) + g (n) = 0$ requires both $f (n) = 0$ and $g (n) = 0$ .

Step 4: Conclude. The only function satisfying this is $f (n) = 0$ for all $n$ , i.e., the zero function itself. So exactly 1 function has an inverse.

Answer: B

Question 5 · Engineering Mathematics MCQ

Match the group with its self-inverse condition.

List I:

P) $Z_{n}$ under addition

Q) $U (n)$ under multiplication

List II:

$x^{2} \equiv 1 (mod n)$
$2 x \equiv 0 (mod n)$

A.
P-2, Q-2
B.
P-1, Q-1
C.
P-2, Q-1
D.
P-1, Q-2

Correct Answer:

Step-by-Step Solution

Key idea: The self-inverse condition depends on the group's operation and identity.

Step 1: For $Z_{n}$ under addition, the identity is 0. The condition is $x + x = 0 ⟹ 2 x \equiv 0 (mod n)$ . This matches 2.

Step 2: For $U (n)$ under multiplication, the identity is 1. The condition is $x \times x = 1 ⟹ x^{2} \equiv 1 (mod n)$ . This matches 1.

Step 3: Therefore, P matches 2, and Q matches 1.

Answer: C

Question 6 · Engineering Mathematics MCQ

Consider the following:

Assertion (A): The number of generators of the cyclic group $Z_{12}$ is 4.

Reason (R): An element $k$ is a generator of $Z_{n}$ if and only if $g cd (k, n) = 1$ .

Which option is correct?

A.
Both A and R are true and R is the correct explanation of A
B.
Both A and R are true but R is NOT the correct explanation of A
C.
A is true but R is false
D.
A is false but R is true

Correct Answer:

Step-by-Step Solution

Key idea: The number of generators of $Z_{n}$ is given by Euler's totient function $ϕ (n)$ .

Step 1: Evaluate Assertion (A). The number of generators of $Z_{12}$ is $ϕ (12)$ .

Step 2: Calculate $ϕ (12) = 12 \times (1 - 1/2) \times (1 - 1/3) = 12 \times (1/2) \times (2/3) = 4$ . So, A is true.

Step 3: Evaluate Reason (R). The condition for $k$ to be a generator of $Z_{n}$ is indeed $g cd (k, n) = 1$ . So, R is true.

Step 4: Check the link. R provides the exact mathematical condition used to compute the count in A. Thus, R is the correct explanation of A.

Answer: A

Question 7 · Engineering Mathematics MCQ

Let $G = Z_{15}$ be the group of integers modulo 15 under addition. How many elements in $G$ have an order of exactly 5?

A.
5
B.
4
C.
8
D.
15

Correct Answer:

Step-by-Step Solution

Key idea: In $Z_{n}$ , the number of elements of order $d$ is $ϕ (d)$ , provided $d$ divides $n$ .

Step 1: Identify $n = 15$ and the target order $d = 5$ .

Step 2: Check if $d$ divides $n$ . Since $5$ divides $15$ , elements of order 5 exist.

Step 3: The number of such elements is given by Euler's totient function $ϕ (d) = ϕ (5)$ .

Step 4: Since 5 is prime, $ϕ (5) = 5 - 1 = 4$ .

Answer: B

Question 8 · Engineering Mathematics MCQ

Let $A = {0, 1, 2, \dots}$ and $F$ be all functions from $A$ to $A$ . Under pointwise addition $(f \oplus g) (n) = f (n) + g (n)$ , which task is IMPOSSIBLE?

A.
Computing $(f \oplus g) (5)$ given $f$ and $g$
B.
Finding $e \in F$ such that $f \oplus e = f$ for all $f$
C.
For arbitrary $f \in F$ , finding $g \in F$ such that $f \oplus g = e$
D.
Checking if $f \oplus (g \oplus h) = (f \oplus g) \oplus h$

Correct Answer:

Step-by-Step Solution

Key idea: Check which task violates the properties of the structure.

Step 1: Evaluate option A. Computing $(f \oplus g) (5) = f (5) + g (5)$ is always possible given $f$ and $g$ . This is possible.

Step 2: Evaluate option B. The identity $e$ satisfies $f (n) + e (n) = f (n)$ , so $e (n) = 0$ for all $n$ . The zero function is in $F$ . This is possible.

Step 3: Evaluate option C. For arbitrary $f$ , finding $g$ such that $f \oplus g = e$ means $f (n) + g (n) = 0$ for all $n$ . If $f (n) > 0$ , then $g (n) = - f (n) < 0$ , which is not in $A$ . This is impossible for most $f$ .

Step 4: Evaluate option D. Since integer addition is associative, $f (n) + (g (n) + h (n)) = (f (n) + g (n)) + h (n)$ for all $n$ . This is possible.

Answer: C

Question 9 · Engineering Mathematics MCQ

Consider the following:

Assertion (A): The operation $*$ defined by $a * b = a + 2 b$ on integers is associative.

Reason (R): For associativity, we need $(a * b) * c = a * (b * c)$ for all $a, b, c$ .

Which option is correct?

A.
Both A and R are true and R is the correct explanation of A
B.
Both A and R are true but R is NOT the correct explanation of A
C.
A is false but R is true
D.
A is true but R is false

Correct Answer:

Step-by-Step Solution

Key idea: Check if $(a * b) * c = a * (b * c)$ using the given formula.

Step 1: Compute $(a * b) * c$ . First, $a * b = a + 2 b$ . Then $(a * b) * c = (a + 2 b) * c = (a + 2 b) + 2 c = a + 2 b + 2 c$ .

Step 2: Compute $a * (b * c)$ . First, $b * c = b + 2 c$ . Then $a * (b * c) = a * (b + 2 c) = a + 2 (b + 2 c) = a + 2 b + 4 c$ .

Step 3: Compare. $a + 2 b + 2 c \neq = a + 2 b + 4 c$ in general (e.g., let $c = 1$ : left gives $a + 2 b + 2$ , right gives $a + 2 b + 4$ ).

Step 4: Evaluate A and R. Since the two expressions are not equal, $*$ is not associative, so A is false. R correctly states the definition of associativity, so R is true.

Answer: C

Question 10 · Engineering Mathematics MCQ

Let $X$ be a set with 4 elements. The group $(2^{X}, Δ)$ is isomorphic to a direct product of copies of $Z_{2}$ . Which of the following is IMPOSSIBLE for this group?

A.
It contains an element of order 4.
B.
It has 16 elements.
C.
It is an Abelian group.
D.
Every element is its own inverse.

Correct Answer:

Step-by-Step Solution

Key idea: Isomorphism to $Z_{2} \times Z_{2} \times Z_{2} \times Z_{2}$ dictates the group's properties.

Step 1: Since $∣ X ∣ = 4$ , the group is isomorphic to $Z_{2}^{4}$ , which has $2^{4} = 16$ elements.

Step 2: In $Z_{2}$ , every non-identity element has order 2. Thus, in $Z_{2}^{4}$ , every element is its own inverse (order 1 or 2).

Step 3: Since $Z_{2}$ is Abelian, the direct product is also Abelian.

Step 4: Because the maximum order of any element is 2, it is impossible to contain an element of order 4.

Answer: A

algebraic structures and groups notes

Chapter Roadmap: Algebraic Structures and Groups

Algebraic Structures and Groups

What is a Binary Operation?

What is a Binary Operation?

The core idea

Formal definition

Custom Operators in Exams

Custom Operators in Exams

Treat the symbol as a machine

Order matters

Try a question

Login to view the complete notes

Personalised first. High quality throughout.

Built around you, not around a syllabus PDF

Revision that hits your weak spots

Questions calibrated to the real exam

Notes written for recall, not for volume

One place for everything

Honest progress

Unlock the whole course

Algebraic Structures and Groups Notes for GATE CS

Chapter Roadmap: Algebraic Structures and Groups

Algebraic Structures and Groups

What is a Binary Operation?

What is a Binary Operation?

The core idea

Formal definition

Custom Operators in Exams

Custom Operators in Exams

Treat the symbol as a machine

Order matters

Commutativity: Does Order Matter?

Commutativity: Does Order Matter?

Definition

Algebraic Structures and Groups: Solved Questions with Step-by-Step Explanations (10 Problems)

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

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